Let $G$ be a finite group. What is a Sylow $p$-subgroup of $G$ ?

Assuming that a Sylow $p$-subgroup $H$ exists, and that the number of conjugates of $H$ is congruent to $1 \bmod p$, prove that all Sylow $p$-subgroups are conjugate. If $n_{p}$ denotes the number of Sylow $p$-subgroups, deduce that

$$n_{p} \equiv 1 \quad \bmod p \quad \text { and } \quad n_{p}|| G \mid$$

If furthermore $G$ is simple prove that either $G=H$ or

$$|G| \mid n_{p} \text { ! }$$

Deduce that a group of order $1,000,000$ cannot be simple.